I just bought X-Wing Alliance (but haven't played it yet), and having designed missions for XvT, I wonder how you create missions (with briefings and all that things) for
Alliance. Do you use the integrated builder
or do you use a "non-official" mission builder? (And if yes, where can I get it?)

The 'intergrated' mission editor is called the 'Skirmish Editor', and is very limited in what you can do.

I recommend AlliED, which is a shareware mission editor that is THE premiere mission utility that exists currently.

Having beta-tested AlliED, and actually still beta testing new versions, AND having tried all other 'discontinued' editors, I have to say that AlliED is very well put together. It's made by Troy Dangerfield, who also made XvtED and X-ED, his two Freeware editors, so you know it's got to be good.